Animation
Theoretical Context
A Gaussian wavepacket is a particle with a wave function \Psi(x,t) that gives a Gaussian distribution for the probability density |\Psi|^2(x,t). The initial wave function can be parametrized as:
\begin{equation*}
\Psi(x,0) = \Big(\frac{2a}{\pi}\Big)^{1/4}\;e^{-ax^2},
\end{equation*}which is properly normalized. Throughout, we set the mass of the particle and the reduced Planck constant to unity, i.e. m=\hbar\equiv 1. The initial wavepacket is stationary in the sense that the whole package does not move in a specific direction. Appending a factor e^{ik_0x} would correspond to a wavepacket moving in the positive x-direction with average momentum k_0.
We take the particle to be free, such that the Hamiltonian is H=p^2/2. Solving for the time evolution of the initial wave function given the free Hamiltonian gives:
\begin{equation*}
\Psi(x,t) = \Big(\frac{2a}{\pi}\Big)^{1/4}\frac{1}{\sqrt{1+2iat}}\;e^{-\frac{ax^2}{1+2iat}},
\end{equation*}for any time t>0. As a function of time, the probability density |\Psi|^2(x,t) is given by:
\begin{equation*}
|\Psi|^2(x,t) = \sqrt{\frac{2}{\pi}}\;w(t)\;e^{-2w^2(t)x^2},
\end{equation*}where w(t)\equiv\sqrt{a/(1+(2at)^2)}. For this distribution, the standard deviation in momentum and position are given by:
\sigma_p = \sqrt{a},\quad\quad\quad \sigma_x(t)=1/2w(t). Physical Interpretation
At t=0, the particle has a relativity well-defined position, since the peak has well defined narrow width. As time increases, observe how the packet spreads out: the location of the particle becomes less confined. This evolution of the wavepacket can be understood based on Heisenberg’s uncertainty principle as follows. Initially, the peak is narrow so that the uncertainty in the position, as measured by \sigma_x, is small. However, if \sigma_x is small, \sigma_p must be large (relative to \sigma_x), in order to satisfy Heisenberg’s uncertainty principle. Physically, this means that at t=0 the momentum (and hence velocity) of the particle is ill-defined. Because of this large uncertainty in the velocity, the uncertainty in position increases on subsequent times as well, causing the wavepacket to spread. The growing uncertainty in position is also evident from the middle panel. The fact that \sigma_p is not time-dependent is due to the fact we considered a free particle here (with V(x)=0), i.e. there is no force acting on the particle. Furthermore, notice that initially the wavepacket saturates the uncertainty principle, i.e. \sigma_x\sigma_p = 1/2, after which the product starts to grow (recall that we set \hbar\equiv 1). This is a key characteristic of Gaussian wavepackets.